#### Date of Award

2020

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy in Industrial and Systems Engineering

#### Department

Mechanical, Industrial and Systems Engineering

#### First Advisor

Manbir S. Sodhi

#### Abstract

The Traveling Salesman Problem (*TSP*), Assignment Problem (*AP*), and Scheduling Problems (*SP*) such as Job-shop or Flow-shop problems are well known in computational mathematics. This dissertation explores four new types of problems that inherit the properties of one or some of the above mentioned problems, each of which will be elaborated as follows:

The first problem considered is the Travelling Salesman Problem with Jobtimes (*TSPJ*) which integrates two factors {distance and job-time in an optimization metric. It is a variant of TSP where the traveler moves through n locations, visiting each location once to initiate one of *n* jobs, and returns to the first location. After initiation of a job, the traveler moves to the next location immediately and the job continues autonomously. The goal is to minimize the time of completion of the last job, i.e. makespan. By using a mathematical model and the heuristics, the results show that the new adapted nearest neighbor algorithm and a proposed local search improvement heuristics solution has less than 6% relative optimality gap with the optimum solution of the sample sets in acceptable processing time. Also, the gaps between the GAMS solutions and the *GA* outputs are less than 10%.

The second problem considered is the Traveling Salesman Problem with Jobtime, Drop-off, and Pick-up (*TSPJDP*). It is a variant of TSP where the traveler/ transporter moves through n given locations, visiting each location first to drop an autonomous agent off to execute a preassigned job, and again to pick the agents up after completion of their job. At the end, the traveler returns to the origin. After initiation of the job at each drop-off location, the traveler can either wait to pick up the agent, or move to another location immediately. The agent continues autonomously and waits to be picked up after the job is completed. The goal is to find the sequence of the drop offs and pick-ups that minimizes the completion time of the entire tour. By using a mathematical model and the heuristics, the results show that the heuristics solutions have at most 3% relative optimality gap with the optimum solution in the sample sets.

The third problem considered is the Multi Circuit Traveling Salesman Problem(MCTSP). It is a variant of TSP where the traveler makes multiple circuits through a given set of nodes, visiting each node once in each circuit without returning to the depot until visiting all nodes at the last circuit. The goal is to find the sequence of nodes that minimizes the tour of all the circuits. By using a mathematical model and the heuristics, the results have at most 9% relative optimality gap with the optimum solution in acceptable processing time in the sample sets. Also, the results show *MCTSP* improves the solutions by an average 3.9% in comparison with the multiple independent *TSP* in the sample sets. However, by increasing the tour length and the number of the nodes, the effect of multi-circuit decreases. In addition, some special properties of the *MCTSP* such as recursive property and its conditions are detailed.

For each of the problems described above, this dissertation uses the following approach to develop solutions based on the properties salient to the problems: 1) It states the problem and illustrates the details by an example; 2) The literature is reviewed for similar problems and the differences are highlighted; 3) Special properties and applications of the problem are detailed; 4) Mathematical models are developed; 5) The mathematical models are programmed with the GAMS language; 6) Heuristics are proposed and programmed in the Python language; 7) The results of the GAMS solver(CPLEX) and the output of the heuristics are analyzed; 8) Variations and their mathematical models are explored; 9) and finally, the results with managerial decisions and a research perspective conclude the dissertation.

Previous research has explored many variations of the *TSP* using distance factor; and studied different variations of the *SP* using job-time factor, independently. However, it is less clear the applications and solutions for the hybrid problems which integrate two factors {distance and job-time. In addition, the previous studies have focused on solving *TSP* in one tour (circuit). However, there are many applications that need a traveler to move through a set of nodes more than once. The present research proposes the hybrid problems involving the properties of both *TSP* and *SP* in multiple circuits. Finally, by focusing on both factors, we provide fresh insights into the Multi-Circuit Traveling Salesman Problem with Job-times and variations, an area of the mathematical models and heuristics that has not been relatively under-studied in the literature.

#### Recommended Citation

Mosayebi, Mohsen, "THE TRAVELING SALESMAN PROBLEM WITH JOB-TIMES, MULTI-CIRCUIT, AND VARIATIONS" (2020). *Open Access Dissertations.* Paper 1205.

https://digitalcommons.uri.edu/oa_diss/1205